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Two step (spherical) square well potential

  1. Oct 1, 2009 #1
    1. The problem statement, all variables and given/known data
    Just need some rough guidance on this one, nothing specific is really needed. The problem:

    Given a spherically symmetric potential (V(r))
    [tex]
    V\left( r \right) = \left\{\begin{gathered}
    V_0 \hfill \hspace{2}r<a \\
    0 \qquad a<r<b \\
    \infty \hfill r>b \\
    \end{gathered} \right
    [/tex]

    Find the energies for the ground state and the first excited state. Also find an (approximate) expression for the energy splitting of the levels if [itex]V_0[/itex] is very large compared to these energy levels.


    2. Relevant equations

    Must use spherical Bessel and Neumann functions.

    3. The attempt at a solution

    The wavefunctions:

    [tex]\mathcal{U}_I(r)=A_rJ_{\ell}(k_1r)[/tex]
    [tex]\mathcal{U}_{II}(r)=C_rJ_{\ell}(k_2r)+D_r n_{\ell}(k_2r)[/tex]
    [tex]\mathcal{U}_{III}(r)=0[/tex]

    [tex]k_I=\frac{\sqrt{2m(V_0-E)}}{\hbar}[/tex]

    [tex]k_{II}=\frac{\sqrt{2mE}}{\hbar}[/tex]

    where the spherical Neumann function ([tex]n_{\ell}[/tex]) goes away in the first function since it is singular at the origin.

    At this point work with the following boundary conditions:

    [tex]\mathcal{U}_I(a)=\mathcal{U}_{II}(a)[/tex]
    [tex]\mathcal{U}_I^{'}(a)=\mathcal{U}_{II}^{'}(a)[/tex]
    [tex]\mathcal{U}_II(b)=0[/tex]

    From here you get a three equations, where the derivatives are taken with respect to K1 and K2 (respectfully). First question: How do I take a derivative of the spherical Bessel and Neumann functions? Should I use the recursion formula and solve for the derivative?

    Actually I am just not quite sure where to go from here in terms of finding the ground state energy. If this were a regular potential (two step) barrier, all I would do is eliminate the three constants using the three equations, arriving at a transcendental equation which I would then graph graph each side of the equation separately and find the intersections which represent the energy eigenvalues. Same thing here? Any thoughts???

    Thanks yall

    IHateMayonnaise
     
  2. jcsd
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